Math 5285H: Fundamental Structures of Algebra I (Fall 2019)

Lectures: MWF 10:10-11:00 in Vincent Hall 113.

Instructor: Gregg Musiker (musiker "at" math.umn.edu)

Office Hours: Mondays 3:00-4:00, Wednesdays 11:00-12:00, Fridays 2:30-3:30 in Vincent 251; also by appointment.

Course Description:

This is the first semester of a course in the basic algebra of groups, ring, fields, and vector spaces. Roughly speaking, the Fall and Spring semesters should divide the topics as follows:

Fall: Vector spaces, linear algebra, group theory, symmetry, and the Sylow Theorems

Spring: Rings, fields, and Galois theory

Prerequisites: Some previous exposure to linear algebra (vectors, matrices, determinants) (such as from 2243, 2373, or 2573) would help. Also, one should either have the ability to write and read mathematical proofs (such as from 2283, 2574, or 3283), or have the desire and drive to learn how.

Required text: Algera, by Michael Artin, (2nd edition 2011, Prentice Hall).

Fall: some (but not all) of Chapters 1-7;

Spring: some (but not all) of Chapters 11,12,13,15,16

Other useful texts (On reserve in math library)
Title Author(s) Year
Introduction to abstract algebra W. Keith Nicholson 2007
Contemporary abstract algebra Joseph A. Gallian 1994
Algebra Thomas Hungerford 1980
Abstract Algebra Dummit and Foote 2004
Abstract algebra: theory and applications Thomas Judson 1994

Grading:

  • Homework (50%): There will be 5 homework assignments due in class as according to the schedule below. For the most part, homework is due every other Friday. The first homework assignment is due on September 20th.

    I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates their collaborators. Late homework will not be accepted. Early homework is fine, and can be left in my mailbox in the School of Math mailroom in Vincent Hall 107. Homework solutions should be well-explained; the grader is told not to give credit for an unsupported answer. Complaints about the grading should be brought to me.

  • Exams (15% each): There will be 2 take-home exams, handed out on October 4th (due October 11th) and November 8th (due November 15th). Each will be open book, open notes, and with calculators allowed. However, for these exams, you are not allowed to consult other electronic sources, such as the internet, and you are not allowed to collaborate or consult with other students or other human sources. These exams are to be collected in class.

  • Final Exam (20%): The final exam will be take-home as well, under the same policies as above, handed out on December 4th, to be turned in during class on December 11th.
  • Class Participation:

    Participation in class is encouraged. Please feel free to stop me and ask questions during lecture. Otherwise, I might stop and ask you questions instead.

    Tentative Lecture Schedule

  • (Sep 4) Lecture 1: Introduction to the Course and Basic Matrix Operations (Artin, Sec. 1.1)
  • (Sep 6) Lecture 2: Row Reductions (Artin, Sec. 1.2)
  • (Sep 9) Lecture 3: Solving Linear Systems of Equations (Artin, Sec. 1.2)
  • (Sep 11) Lecture 4: Lagrange Interpolation and Uniqueness of the Determinant (Artin, Sec. 1.2, 1.4)
  • (Sep 13) Lecture 5: More on Determinants and Transposes (Artin, Sec. 1.3-1.4)
  • (Sep 16) Lecture 6: Permutations and Other formulas for the Determinant (Artin, Sec. 1.5-1.6)
  • (Sep 18) Lecture 7: Introduction to Groups and Subgroups (Artin, Sec. 2.1-2.2)
  • (Sep 20) Lecture 8: More on Groups and Subgroups (Artin, Sec. 2.1-2.2)
  • (Sep 23) Lecture 9: Subgroups of the Integers (Artin, Sec. 2.3)
  • (Sep 25) Lecture 10: Cyclic Groups (Artin, Sec. 2.4)
  • (Sep 27) Lecture 11: Homomorphisms (Artin, Sec. 2.5)
  • (Sep 30) Lecture 12: Isomorphisms and Normal Subgroups (Artin, Sec. 2.6, 2.8)
  • (Oct 2) Lecture 13: More on Automorphisms (Artin Sec. 2.6, 2.8)
  • (Oct 4) Lecture 14: Equivalence Relations, Cosets and Modular Arithmetic (Artin, Sec. 2.7-2.9)
  • (Oct 7) Lecture 15: Applications of Lagrange's Theorem (Artin, Sec. 2.8)
  • (Oct 9) Lecture 16: Quotient Groups and Noether's First Isomorphism Theorem (Artin, Sec. 2.12)
  • (Oct 11) Lecture 17: Product Groups and the Correspondence Theorem (Artin, Sec. 2.10-2.11)
  • (Oct 14) Lecture 18: Review of Vector Spaces and Introduction to Fields (Artin, Sec. 3.1-3.2)
  • (Oct 16) Lecture 19: More on Finite Fields (Artin, Sec. 3.2)
  • (Oct 18) Lecture 20: Constructing F4 and Abstract Vector Spaces (Artin, Sec. 3.3)
  • (Oct 21) Lecture 21: Vector Spaces Isomorphisms and Bases (Artin, Sec. 3.4)
  • (Oct 23) Lecture 22: Dimension of Vector Spaces and Direct Sums (Artin, Sec. 3.5-3.6)
  • (Oct 25) Lecture 23: More on the theory of Direct Sums (Artin, Sec. 3.5-3.6)
  • (Oct 28) Lecture 24: Infinite Dimensional Vectors Spaces and Beginning Linear Transformations (Artin, Sec. 3.6-3.7, 4.1-4.2)
  • (Oct 30) Lecture 25: The Dimension Formula and More on Linear Operators (Artin, Sec. 4.1-4.3)
  • (Nov 1) Lecture 26: Eigenvectors and the Characteristic Polynomial (Artin, Sec. 4.4-4.5)
  • (Nov 4) Lecture 27: Triangular and Diagonal Forms (Artin, Sec. 4.6)
  • (Nov 6) Lecture 28: Jordan Canonical Form (Artin, Sec. 4.7)
  • (Nov 8) Lecture 29: Orthogonal Matrices (Artin, Sec. 5.1)
  • (Nov 11) Lecture 30: SO_2, SO_3, and Introduction to Bilinear Forms (Artin, Sec. 5.1 and 8.1-8.2)
  • (Nov 13) Lecture 31: Symmetric Forms, Hermitian Forms, and the Return of Orthogonality (Artin, Sec. 8.2-8.4)
  • (Nov 15) Lecture 32: The Spectral Theorem (Artin, Sec. 8.4-8.6)
  • (Nov 18) Lecture 33: Isometries (Artin Sec. 6.1-6.2)
  • (Nov 20) Lecture 34: Isometries of the Plane (Artin, Sec. 6.3)
  • (Nov 22) Lecture 35: Finite Subgroups of Isometries of the Plane (Artin, Sec. 6.4)
  • (Nov 25) Lecture 36: Abstract Symmetry: Group Operations (Artin, Sec. 6.7)
  • (Nov 27) University Classes Cancelled Due to Snow - Have an Early Thanksgiving!
  • (Nov 29) Happy Thanksgiving!
  • (Dec 2) Lecture 37: Actions on Cosets, Subsets, and the Counting Formula (Artin, Sec. 6.8-6.10)
  • (Dec 4) Lecture 38: Class Equation (Artin, Sec. 7.2)
  • (Dec 6) Lecture 39: More on the Class Equation, Permutation Representations, and p-groups (Artin, Sec. 6.11, 7.1, 7.2, 7.3)

  • Homework assignments and exams
    Assignment or Exam Due date Problems from Artin text,
    unless otherwise specified
    Homework 1 Friday 9/20 Chapter 1: # 1.2, 1.6, 1.7, 1.10, 1.11, 1.13, 2.2, 2.5, 2.6, 2.10, 4.1, 4.6, 5.1, 5.3, 6.2, M.7
    Chapter 2: # 1.3, 2.1, 2.2, 2.4
    Homework 2 Friday 10/4 Chapter 2: # 2.5, 3.1, 4.1, 4.2, 4.3, 4.7, 4.8, 4.9, 5.1, 5.2, 5.3, 5.4, 5.5, 6.3, 6.10, 6.11
    Exam 1 Friday 10/11 Take Home Exam available at Canvas
    Homework 3 Friday 10/25 Chapter 2: # 8.1, 8.3, 8.4, 8.6, 8.7, 12.1, 12.2, 12.4, 12.5
    Chapter 3: 1.1, 1.2, 1.3, 1.4, 1.5, 1.8, 1.9
    Homework 4 Friday 11/8 Chapter 3 (Wrote out beginning of each problem in case of different numbering in your book):
    1.11 (Prove that the set of symbols {a+bi| ...}),
    2.2 (Which of the following subsets is a subspace ...),
    5.1 (Prove that the space R^(nxn) of all nxn real matrices ...),
    5.2 (The trace of a square matrix ...)

    Chapter 4: 2.3, 3.2, 4.1, 4.2, 4.3, 4.4, 4.6, 5.1, 5.3, 6.1, 6.3
    Exam 2 Friday 11/15 Take Home Exam available at Canvas
    Homework 5 Wednesday 12/4 Chapter 5: # 1.3, 1.5
    Chapter 6: # 3.2, 4.1, 4.2, 4.3, 5.1, 7.1, 7.3, 11.1
    Chapter 8: # 1.1, 3.1, 3.2, 3.3, 3.4, 6.3, 6.17*, 6.18, M.1

    Special Office Hours: Tuesday December 3rd, 2:00-3:00 pm
    Final Exam Wednesday 12/11


    Class Policies:

  • S/N Grade: If you are registered S/N, I will submit a grade of S if your letter grade is C or above, and otherwise a grade of N.

  • Incomplete grade: A grade of 'I' will only be considered when failure to complete all course requirements is for reasons beyond the student's control. The minimum requirement for an incomplete grade is a substantial amount of course work completed at the level of C or better. An 'I' grade also requires a written agreement between the student and the instructor on how the missing work will be completed. After one year, an 'I' turns into an 'F' if the course work is not completed. Any arrangement for an incomplete grade can only be considered before the final exam.

  • Disability Accommodations: Disability services promotes access and equity for everyone at the U of M. If you are registered with DS and would like to discuss accommodations, please contant the instructor as soon as possible. If you require accommodations, but are not registered with DS, please contact their office.

  • To add or drop the course: For the various rules and deadlines for adding or dropping the course, visit One-Stop.

  • Scholastic Misconduct: Academic dishonesty in any work for this course will be grounds for awarding a grade of 'F' for the entire course. University policies regarding academic dishonesty, credit, workload expectations, and grading standards are available here and here.